### Abstract

We study comparator networks for selection, merging, and sorting that output the correct result with high probability, given a random input permutation. We prove tight bounds, up to constant factors, on the size and depth of probabilistic (n, k)-selection networks. In the case of (n, n/2)-selection, our result gives a somewhat surprising bound of Θ(n log log n) on the size of networks of success probability in [δ, 1 - 1/poly (n)], where δ is an arbitrarily small positive constant, thus comparing favorably with the best previously known solutions, which have size Θ(n log n). We also prove tight bounds, up to lower-order terms, on the size and depth of probabilistic merging networks of success probability in [δ, 1 - 1/poly(n)], where δ is an arbitrarily small positive constant. Finally, we describe two fairly simple probabilistic sorting networks of success probability at least 1 - 1/poly(n) and nearly logarithmic depth.

Original language | English (US) |
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Pages (from-to) | 559-582 |

Number of pages | 24 |

Journal | Theory of Computing Systems |

Volume | 30 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 1997 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*Theory of Computing Systems*,

*30*(6), 559-582. https://doi.org/10.1007/s002240000068